This paper is organized as follows. In section , we define operators size and complexity for discrete systems. In section , we prove that if the metric on the unitary group is a function of operator size, then typical sectional curvatures are always positive. In section , we analyze the sectional curvatures distribution, and illustrate that 2-surfaces generated by Hamiltonians of size smaller than the typical size can have negative curvatures. The requirement that sectional curvatures should be at the order of \(1/N^2\)  constrains the form of the complexity metric up to constant corrections. In section , state-dependent operator size and state complexity are defined. The curvatures of 2-surfaces generated by Hamiltonians of size munch smaller than \(N/2\)  is negative. Conclusions are in section . In appendix , results relevant to sectional curvatures in [1]} are briefly reviewed. In appendix , the distribution of sectional curvatures is derived.
